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					Diaspora of a Mathematics of Argument

                   R. Loui
          Dept of Computer Science
           Washington University
                  St. Louis

I. Intellectual History of Process-Based Models of Reasoning
II. Some Technical Issues regarding Argument
III. Foundations:
   A. Probability
   B. Decision
   C. Legal Reasoning
   D. Belief Revision/Deontic Logic
   E. Negotiation
   F. Rhetoric
IV. Future Work
   A. Fairness
   B. Computation
                          CS TALK
                 Scope of My Current CS Work

cgi in gawk
    book with S. Sachs

independent co-malloc for localizing dynamically allocated objects
optimal average hash chain length for gawk
malloc with a vmstat time series estimator for elective memory expansion
   gnu release(s) with M. Waldvogel, M. Pachos, K. Krouse

something with FPGA's
   patent & license with J. Lockwood, J. Moscola, M. Pachos
                      CS TALK
          Scope of My Current CS Work (cont.)

purely probabilistic negotiating agents
   model, simulations

real-time object recognition for aerial targets
                                     hiding among non-combatants
   half-baked ideas, proposal with R. Pless

AI and Law service
   journal, ICAIL, JURIX, special issues, workshops, JD-PhD's, no $
                  WHAT IS LOGIC?
        What do Computer Scientists think is Logic?
Roughly: Hilbert-Russell-Whitehead tradition:

   1. there is one correct logic:
         it is either the predicate logic or the propositional logic or both

   2. entailment (syntactic or semantic?) has something to do with
        mathematical proof

   3. logic codifies correct ways of reasoning

   4. logic has something to do with the success of hardware
               WHAT IS LOGIC?
     What do Computer Scientists think is Logic?
Some more advanced members of our species:
                      Knowledge Representation

  1. logics are like programming languages; can be chosen or designed
       w/o metaphysical consequence

  2. some logics are more expressive than others

  3. some logics license more inferences than others

  4. inferential license and expressiveness are complementary
   What is the Difference between Argument and
Diagram of an argument
      <p, { < {a,b}, p >,
                     <{c}, a>,
                     <{d}, b>

Diagram of a proof
      <p1, p2, …, p>

Argument                     Deduction

nondemonstrative             demonstrative
if p then defeasibly q       if p then materially q
nonmonotonic                 monotonic
argument                     proof
subargument                  subproof
counterargument              (counterproof?)
defeat                       (fallibility? corrigibility?)
inconsistency-tolerating     inconsistency-degenerating
                              Less Obvious
Argument                           Deduction
focus is on metalanguage:          focus is on object language:
conflict, rebuttal, warrant        and, or, not

anytime:                           ideal
warrant w.r.t. arguments           commitment at all t
produced in time t

Warranted(S,t)                     Thm(S), Proved(S,t)

constructive:                      nonconstructive
p's warrant underdetermined

ampliative: nondeterminism         nonampliative: conclusions
of process                         in the meanings of words
                    Less Obvious (continued)
Argument                           Deduction

strategy-based:                    constraint-based:
choices                            proof = constraint-propagation

dialectical                        unilateral

sits between objectivist and       invites principle of charity:
relativist conceptions of truth

formally, 10-20 years old          formally, 100-150 years old

20thC seminal in social sciences   20thC dominant in logic
                    Variations of Logic
Modal Logic         opaque contexts
                    >> notation for beliefs about beliefs

Fuzzy Logic         predication weakened
                    >> smoother control, washing machines

Multivalued Logic   truth weakened
                    >> semantic curiosities, reductions

Relevance Logic     implication weakened
                    >> model of limited inferential capacity
                     Variations of Logic (cont.)
Intuitionist Logic         weak negation added
                           >> first step toward elevation of process

Counterfactual Logic       second implication added
                           >> plausible alternative conditional

Paraconsistent Logic       meaning from inconsistency
                           >> proof-theory for consistent subsets

Belief Revision            recovery from inconsistency
                           >> model of premise adoption/retraction

Argument                   ties logic to computation in a fundamental way
                           >> rewrite foundations of other fields

1. where did the idea of defeasibility come from?
2. where did the idea of procedural rationality come from?
3. where did the idea of argument come from?
    "tweety is a bird, but tweety does not fly"
                 McCarthy-Hayes -- Modal Belief
                            Reiter --Closed World Databases
                            Doyle -- TMS
                            Kowalski -- PROLOG
                            Clark -- Negation as Failure
                                                  Argument systems
      Pollock -- Defeasible Reasoning
                            Nute -- Defeasible PROLOG
Kyburg -- system for probability based on defeat
       "it seems red therefore it is red" is defeasible
                                                             Belzer -- Defeasible Reasoning
                                                   Swain -- Epistemic Defeasibility
                                        Pollock -- Knowledge & Justification
                  Sosa -- Conceptions of Knowledge
                  Lehrer-Paxson -- Knowlege
                  Firth -- Coherence
       Chisholm -- Perceiving
Ladd -- Structure of a Moral Code
      Reasoning (Qualitative Decision Theory):
"doing a achieves the goal, therefore do a" is defeasible

                       Nozick -- Practical Reason/Explanations
                        Searle -- Prima Facie Obligations/Practical Reason
    Raz -- Practical Reason/Norms
 Gauthier -- Practical Reasoning
                   Ethical Reasoning:
"a person has a prima facie obligation or responsibility"
                                 Glover -- Responsibility
                                 Nozick -- Moral Structures
                      Feinberg -- Action and Responsibility
            Wellman -- Language of Ethics
    Brandt -- Blameworthiness and obligation
    Melden -- Action/Rights
    Mackie -- Responsibility and Language
 Hare -- Language of Morals

 (note: Stevenson 1938 and Ross 1930)
              Political Justification:

Barry -- Political Argument
Rawls -- Pure Procedural Justice

                   Rescher -- Dialectics
         Perelman -- Justice and Argument
Toulmin -- Uses of Argument
                         Origins (continued):
                    Hart -- Ascription of Responsibility
         Wisdom -- Gods
         Waismann -- Verifiability
         Austin -- Speech Acts
   Wittgenstein -- Remarks on Foundations of Mathematics
Keynes -- Treatise on Probability
1908 thesis begins “Part of our knowledge we obtain direct; and part by argument.”
Bentham -- Principles of Morals and Legislation
When the student has learnt that in English law here
   are positive conditions required for the existence of
   a valid contract, - his understanding of the legal
   concept of a contract is still incomplete, ... For
   ... he has still to learn what can defeat a claim
   that there is a valid contract, even though all these
   conditions are satisfied. The student has still to
   learn what can follow on the word "unless", which
   should accompany the statement of these conditions.
   This characteristic of legal concepts is one for
   which no word exists in ordinary English ... but the
   law has a word which with some hesitation I borrow
   and extend: this is the word “defeasible”...

(Hart vs. Aristoteliean Society, 1951, p. 152)
... [Principia Mathematica] gives rise to questions
   about the relation in which ordinary reasoning stands
   to this ordered system, and in particular, as to the
   precise connection between the process of inference,
   in which the older logicians were principally
   interested, but which [Russell] ignores, and the
   relation of implication, on which his scheme depends.

The gradual perfection of the formal treatment ... had
  been to empty [logic] of content and to reduce it
  more and more to mere dry bones, until finally it
  seemed to exclude ... most of the principles, usually
  deemed logical, of reasonable thought.

(Keynes vs. Russell, Whitehead, Ramsey, 1908/1921/1973,
   p. 118, 1972, p. 243)
                Formal Inconsistency:
WITTGENSTEIN: Think of the case of the Liar. It is
   very queer in a way that this should have puzzled
   anyone-- Because the thing works like this: if a man
   says 'I am lying' we say that it follows that he is
   not lying, from which it follows that he is lying and
   so on. Well, so what? ... It does not matter. ...
   it is just a useless language-game, and why should
   anyone be excited?

TURING: ... one usually uses a contradiction as a
   criterion for having done something wrong. But in
   this case one cannot find anything done wrong.

WITTGENSTEIN: Yes -- and more: nothing has been done
   wrong. ... where will the harm come?

(Wittgenstein vs. Turing, 1939, Hodges, p. 154)

0. knee-jerk reaction (deductivists)
   Q. isn't
                  p > r,
                  p & q > ~r
                           reducible to
                  p & ~q --> r
                  p & q --> ~r       ?

   A. no.
                  what can be concluded with just p? r.
                  does that imply ~q? no.
I. old issues (Touretzky, Horty, Thomason, 1987)
    Q. skeptical vs. credulous
          A. skeptical
    Q. ambiguity-propagating vs. blocking
          A. depends whether undercut or rebut
    Q. syntactic specificity
          1. (strict specificity)
                    p>r                            p & q > ~r

        2. (shortcut specificity)
                 p > q; q > r                      p > ~r

        3. (defeasible specificity)
                 p>r                               q > ~r
           TECHNICAL (continued)

A.   keep it simple (Nute, 1990)
A.   appeal to convention (Simari-Loui, 1992)
A.   give explicit ordering (Lin-Shoham, 1987, Vreeswijk, 1991)
A.   provide for meta-argument about defeat
          e.g., context-dependent defeat (Prakken-Sartor, 1995)

                  [r1] p > q
                  [r2] r > ~q
                  s > [r1>r2]
                  TECHNICAL (continued)

II. principles versus rules (Loui-Norman, Hage, Verheij, 1995-2001)

   Q. Is there a formal difference between:

        1. no vehicles in the park

        2. parks should be peaceful

   A. Rationales: can recall rationales during dispute
   A. Principles can be weighed: free speech versus privacy
                  TECHNICAL (continued)

III. rules for fair dialectic (Loui, Gordon, Vreeswijk, Lodder, 1992-2001)

   Q. What is the exact procedural burden?
       1. pro: argument 1 for p
       2. con: argument 2 for ~p, and
                 argument 2 defeats argument 1.

         a. should con have the burden of showing
                 argument 2 defeats argument 1?
         b. should the claim be presumable and subject to dispute?
               TECHNICAL (continued)
Q. are normal default rules fair?
     1. pro: argument 1 for p
     2. con: a. proof of ~q
             b. (it does not suffice to argue ~q?)

Q. what is the penalty of failed attempts to rebut?
     1. pro: argument 1 for p based on b and c, etc.
     2. con: ~b and ~c.
     3. pro: why ~b?
     4-15. con: ... pro: ... con loses the subdispute over ~b.
     16. con: nevertheless, ~c.
A. rhetorical costs: HIGH. logical costs: NONE?
                  TECHNICAL (continued)
IV. rules extracted from cases
   Q. What is the structure of a precedent case?
         A. (Raz, 1970)

        A. (Rissland-Ashley, 1985-1990)
                a+ b+ c+ d- e- f- / q+

        A. (Loui-Norman, 1992)
                argument 1(a,(b,c); q)
                argument 2(a,d,e; ~q)
                argument 3(e,f; ~d)
                argument 4(c; ~f)
               TECHNICAL (continued)
Q. What is the rule of the case?
    A. (Loui-Norman, 1995)
              sufficient premises of arguments in
              dialectical subtree with leafs that are pro arguments

              but no false specificity:

     A. (Prakken-Sartor, Bench-Capon, 1992-2001)
             {Argument1, Argument3} > {Argument2, Argument4}
                  TECHNICAL (continued)

V. criteria for theory-formation when theories are defeasible? (Peczenik,
   McCarty, 1997-2001)

   Q. Given a set of cases:
       case 1 a b d e f j           q
       case 2 a b d                 ~q
       case n b d                   ~r

        what is the "best-fitting" set of defeasible rules?

   1. all cases predicted by rules
   2. no error
         (so far this is a learning problem with no simplicity measure)
   3. sets of rules restricted or justified by principles?
             FOUNDATIONS: I. Probability
A probability calculation is an argument.
A statistical argument is an argument.

Reference Class:

        Reichenbach (1949): "use the narrowest reference class for
   which there are adequate statistics"
        Kyburg (1961,1974,1983): maximum in a partial order?
   dominance ~= defeat. Each "inference structure" permits an argument
   from a different sample class.

         Prob(A | B C D)?
                  Sample from BCD: 5 A's/9
                  Sample from BC: 14 A's/20
         what is the logic of combinatorial significance tests?
FOUNDATIONS: I. Probability (cont.)

<S1,[p1,q1]> and <S2,[p2,q2]> disagree
not([p1,q1] in [p2,q2]) and
not([p2,q2] in [p1,q1])

<S1,[p1,q1]> defeats <S2,[p2,q2]>
they disagree and
S1 strict subset of S2

also Pollock (1985, 1990) who uses defeat explicitly
     FOUNDATIONS: I. Probability (cont.)

     Shafer on Monte-Hall type probability "paradoxes" (1985)

             the probability argument
             is improved through knowledge of the protocol

     Neyman-Pearson tradition of crucial tests

             two crucial tests
             produce two competing statistical arguments?
          FOUNDATIONS: II. Decision
Problem of Small Worlds

     Savage (1954, 1967):
             considering fresh and stale
             should not change the calculation
             based on good and rotten.
                               But of course it does.
             So: a grand world which contains all detail.
                               pseudomicrocosm vs. real microcosm.

     Shafer-Tversky (1988):
             framing problems
             constructive decision theory
      FOUNDATIONS: II. Decision (cont.)

Loui (1990):

     u(s) given T(P1,s)          m(P1) = 5
     u(s) given T(P2,s)          m(P2) = -4
     u(s) given T(P1 & P2, s) m(P1 & P2) = 6; u(s) = 5-4+6
              ("defeasibility" of linearity)

     Holds(P1&P2, s) > u(s) = 7
     Holds(P1,s) > u(s) = 5
             (defeasibility of valuation)
      FOUNDATIONS: II. Decision (cont.)

But s is a lottery: s = {r/p; t;(1-p)}
Prob(p) = .5; u(r) = 10; u(t) = 0; so expected u(s) = .5(10) +.5(0) = 5
                (defeasibility of outcome)

Simon (1955-1967)
    substantive vs. procedural rationality, yes,
              but more importantly:
    decision is more like chess than constraint-propagation:
    heuristic valuation changes as search/computation proceeds

a defeasible independence/substitution axiom?

paradoxes of certainty, menu-dependence, framing effects based on
FOUNDATIONS: III. Legal Reasoning
FOUNDATIONS: IV. Belief Revision/Deontic Logic
  contrary-to-duty imperatives (Chisholm, 1963, Nute etc., 1996)

        1. O(p-->q), O(p-->r) and O(~p) are consistent.
                (there can be two expiations)
        2. O(~p) entails O(p --> x) for any x.
                (all expiations are obliged)

                               von Wright (1982):
     "It only means that, if the prohibition is violated, the coordinated
   Contrary-to-Duty imperatives require, for their satisfaction, that both q
   and that r come true. ... If ... the conjunction of the two ... is a logical
   impossibility ..., the legislator would presumably take steps to remove
                                   the conflict."
        "deontic obligation" is different from "technical obligation"
FOUNDATIONS: IV. Belief Revision/Deontic Logic

         3. O(A|B), O(~A|C) entails ~(B&C)

                                 Alchourron (1993):
       "a set of conditional general norms entails ... a non-tautological
   sentence ... iff it follows in the logic for normative propositions that the
          authority has inconsistently normed some action for some

If norms are defeasible rules, no such problems:
    1. two different arguments for expiation.
    2. material conditionals are NOT deontic conditionals.
    3. the entailment is not a result for defeasible conditionals
FOUNDATIONS: IV. Belief Revision/Deontic Logic
   AGM belief revision: choice and refinement, ampliativity
      if p then defeasibly q
      if p and r then defeasibly ~q

       p --> q in K0
       p & ~r --> q in K0[p&r-->~q][r]

                          Alchourron (1993):
     "It seems to me unquestionable that the main [conditions] are the
    formal representation of the revisions effectively performed by an
                  agent and of his dispositions to revise."
   "The particular details of the revisions (and the choice functions) are
      never analyzed by a logician (as a logician)..." Yes, defeasible
    conditionals would inform choice functions, but invoke "possible
       confusion of logic and revision," hiding "conceptually weaker
                     conclusions" in "quiet darkness."
            FOUNDATIONS: V. Negotiation

Acceptability can be argued:
  Fisher-Ury-Patton (1981): Principled negotiation gives arguments for

         “why not open the window?” “I’m cold” “I have a sweater”

   Sycara (1988-1995), Loui-Moore (1993-1997), Parsons-Jennings etc.
   (1998-2001): case-based arguments from precedent settlements

         “that raise was acceptable to you last year”

Utility can be argued:
   search can lift utilities at the proposed settlements
            FOUNDATIONS: V. Negotiation

Utility can be argued:
   Instead of strategic form {a1, …, a4} X {b1, …, b5} with
   utilities Ua and Ub,
          OPTa(x,y) and OPTb(x,y), a hard optimization problem for
          each agent with parameters determined by the agreement

   Ua(x,y) is a’s current best solution for OPTa(x,y)

   Ua and Ub are lifted at <p,q> which is the focus of dialogue
        or when joint-problem-solving

   Sunk cost-of-search arguments lead to settlement
FOUNDATIONS: VI. Rhetoric/"Informal Logic"
           Future Work: Fairness (Procedural)
Claim 1. Fairness depends on the computational abilities of agents (the
   known subspace S x T of the possible strategies S* x T*;

   thus, rules are changed when a strategy s+ is discovered for which all
   known responses t are inadequate.

Claim 2. To be fair (just), the procedure must construct its output upon
   the right inputs, with adequate monotonicity and invariance properties;

   thus, the justification of social procedure resembles the proof of
   program correctness.

   (concatenation) ex-post asymmetry of position should be the result of
   fair (just) ex-ante asymmetry adjusted only by the procedure’s effects
   on elective inputs (strategic choices).
                Future Work: Fairness (cont.)
Obsv (political scientists).
  the purposes of the procedure can limit the degree of stochastics, the
  maximum variation of outcome, and the permissible input types.

Claim 3.
   procedures should be non-dictatorial (for every important different kind
   of outcome, e.g., victory/defeat, there is a strategy pair that would reach
   this outcome). (dominance is more interesting…)

  rule symmetry and equivalent initial position are prima facie fair
  (but sometimes there are good reasons for bias, e.g. plaintiff)

Claim 4.
   Fairness can be inherited from class relationships among procedural
                 Future Work: Computation
Observations. Social procedures which regulate/distribute/construct
  distributions are games. Social programming is like distributed
  programming (quantify over strategy tuples). Building societies is like
  inventing algorithms for distributed decision-making. Argument
  games. Welfare distributions. Elections. Tournaments.

Objective. I want computer science to be at the foundation of the study of
  social procedures.

Obstacle. Game-playing is not considered computation (yet).

Claim. Two people playing chess compute the outcome of the game.

        Why do you have trouble with this claim? But not modems or
   chess tournaments for charities.
                Future Work: Computation
Paradigmatic computation:
        on a machine:           but, long division by hand?
        causally connected:     but, two people doing long division?
        deterministic:          but, probabilistic algorithms?
        locus of control:       but, distributed algorithms?
        fully specified:        but, pseudo-code? uncompiled code?
        algorithmic:            but, protocols? interactions?
                                (control systems?)
        non-elective:           but, frequently arbitrary choices
                                (e.g., search algorithms)

             Protocol-design is a kind of programming.
                 Future Work: Computation

Broad computation: intentional and teleological rule-following upon

                      The existence of a program
                        is the test for computation,
                     not the existence of an algorithm.

Not just any rule-following is computation, since the objects must be
  symbolic and the rule-following purposive and non-accidental.